Out of curiosity, where are things like muted users, followers and relay lists stored? Are they client-dependent?

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They are saved in the relays, on the other hand there is no follower counter, but it is added each time by asking several relays

I see. So if I'm connected to 3 relays, and I mute someone, that information is updated on those three relays. So what if the after that I add 3 new relays? Is it slowly propagated, or do I have to mute the user again? Is there a way to check who I've muted, to "manage" the list?

You propagate the information, no comunication between relays. Yes a mute list is a standard nostr list (NIP-51).

You can manage here for example.

https://listr.lol/

Sweet, thanks!